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Cofinitely Hopfian groups, open mappings and knot complements

A group $Γ$ is defined to be cofinitely Hopfian if every homomorphism $Γ\toΓ$ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is cofinitely Hopfian if and only if it has trivial centre. Applications to the theory of open mappings between manifolds are presented.